Balanced diagonal conjecture for frequency squares

Let $L$ be a frequency square of type $F(m\lambda;\lambda^m)$. A balanced diagonal conjecture asserts that if $(m-1)\lambda$ is even, then $L$ contains a balanced diagonal.

The paper establishes this for $m\in\{1,2,3\}$ and gives constructions and observations supporting the claim for $m\geqslant4$.